Visible crew/equipment: When Selena visits the school, there's a tube reflected on her window. When her car starts to move we see it's part of an umbrella-like structure used for lighting.
Revealing mistake: When Supergirl flies through the window to fight the invisible monster, half of the screen shows her jumping in a school uniform and the other half exiting with the Supergirl outfit. Watch closely and you'll notice that her halves don't match, giving the sense that she is missing part of her body.
Continuity mistake: When Selena holds on to the fridge door an American flag appears and disappears between shots.
Continuity mistake: Supergirl lands the colliding car in a shadowy place but a couple seconds later the shadow is completely gone.
Continuity mistake: The mirror in the castle changes to transparent glass when Supergirl flies through it. After the vortex swallows Selena and Bianca, it turns back into a mirror.
Revealing mistake: In the castle, while Bianca says "I think I'll be going", the glass on the mirror has wrinkles, revealing it's a piece of aluminium foil.
Continuity mistake: Selena parks her car close to the entrance, but in the next shot the car has moved 12 feet to the right.
Continuity mistake: When Selena's car arrives at the campus she is not in the car. A shot later she appears inside.
Continuity mistake: Inside Selena's car, the first shot of the Omegahedron shows her thighs and the book on her lap. In the following shot Selena and the book vanish.
Revealing mistake: After the bulldozer destroys the gas station, two cars crash against each other. Their hoods are already damaged before the crash.
Revealing mistake: Right after Supergirl saves Lucy in the bulldozer and dismounts its front part, when she is about to land in the yard to save Ethan, wires holding her are visible.
Revealing mistake: During the beach scene, when Supergirl flies in front of Ethan, the wires holding her are visible.
Answer: The Omegaahedron is the power source for the whole city. It supplies the air, the dome that protects them, the lights, and the machines they need to survive.